By means of Surface Evolver, a software package of brute-force energy minimization over a triangulated surface developed by the geometry center of the University of Minnesota [Exp. Math 1, 141 (1992)], we have numerically searched the nonaxisymmetric shapes under the Helfrich spontaneous curvature (SC) energy model. We show that there are abundant mechanically stable nonaxisymmetric vesicles in the SC model, including both regular shapes with intrinsic geometric symmetry and complex irregular ones. We report in this paper a catalog of interesting shapes including a corniculate shape with six corns, a quadriconcave shape, a shape resembling sickle cells, a shape resembling acanthocytes, and two tubelike shapes. Most of these shapes can be related to experimental observations in red blood cells and other experiments in fluid membrane. [S1063-651X(98)06510-6].